I personally would not shower with the silver necklace on or any other metal jewelry for that matter.
There is a possibility of the silver necklace becoming tarnished as a result of the water and the chlorine and other chemicals in the water.
The metal itself of the silver necklace won't be harmed when showering with the silver necklace on but you risk tarnishing it so that's one reason why I would not advise showering with the silver necklace on.
It's not worth the risk of damaging the finish of the silver necklace so I would take all metal jewelry off before showering, swimming etc.